Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C22×C4 |
| direct product | C9×#1979 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | #121 |
| commutator | C2×C4 | C22×C4×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle | C22×C3 | C3×#121 |
| prime radical | C23×C3 | C3×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C22×C4×C9 ◁ G | |
414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5,6,7,8,9)(11,13)(14,15,19,24,20,23,18,32)(16,21,27,36,26,37,33,35)(17,28,30,38,31,42,29,22)(25,39,41,34,40,44,43,45);(10,11,12,13)(14,16,18,27,20,26,19,33)(15,21,24,35,23,37,32,36)(17,25,29,41,31,40,30,43)(22,34,28,39,38,45,42,44);(14,17,19,30,20,31,18,29)(15,22,32,42,23,38,24,28)(16,25,33,43,26,40,27,41)(21,34,36,44,37,45,35,39)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 711);